International Journal of Structural Integrity
Impact of plasticity generated by Rayleigh waves on the residual stress behavior
of structural components subjected to laser peening
Anoop Vasu Ramana V. Grandhi
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Anoop Vasu Ramana V. Grandhi , (2015),"Impact of plasticity generated by Rayleigh waves on the
residual stress behavior of structural components subjected to laser peening", International Journal of
Structural Integrity, Vol. 6 Iss 1 pp. 107 - 123
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Impact of plasticity generated by
Rayleigh waves on the residual
stress behavior of structural
components subjected to
laser peening
Plasticity
generated by
Rayleigh
waves
107
Received 11 June 2014
Revised 11 June 2014
Accepted 2 September 2014
Downloaded by University of Louisville At 10:01 05 February 2015 (PT)
Anoop Vasu and Ramana V. Grandhi
Department of Mechanical and Materials Engineering,
Wright State University, Dayton, Ohio, USA
Abstract
Purpose – The impact of laser peening on curved geometries is not fully comprehended. The purpose
of this paper is to explain the action of laser peening on curved components (concave and convex
shapes for cylindrical and spherical geometries) by means of shock wave mechanics.
Design/methodology/approach – An analytical formulation is derived based on the plasticity
incurred inside the material and the results are compared with the prediction by numerical simulation.
Findings – A near-linear relationship is observed between curvature and compressive residual stress;
an increasing trend was observed for concave models and a decreasing trend was observed for convex
models. The consistency in the analytical formulation with the simulation model indicates the behavior
of laser peening for curved geometries.
Originality/value – The differences observed in the residual stresses for spherical and cylindrical
geometries are primarily due to the effect of Rayleigh waves. This paper illustrates the importance of
understanding the physics behind laser peening of curved geometries.
Keywords Curved geometry, Finite element analysis, Laser peening, Reduced plasticity,
Residual stress
Paper type Research paper
1. Introduction
The presence of high tensile stresses on structural components is a potential reason
for cracks to appear and eventually cause the failure of the components. Surface
enhancement techniques, such as laser peening, are usually used to neutralize the
tensile stresses by inducing compressive residual stresses (CRS); they are widely
applied in various fields such as aerospace, automotive, medical, manufacturing, and
nuclear industries. Leap et al. (2011) evaluated the effect of laser peening for improving
fatigue resistance in a severe arrestment hook shank application for a carrier-based
naval aircraft. They found substantial improvement in the crack initiation life of laser
peened specimens in comparison with shot peened specimens. Ganesh et al. (2012)
assessed the performance of laser peening as a possible alternative for the existing shot
peening process for enhancing the life of leaf springs. They attributed the significant
increase in life in the case of laser peening (in comparison with shot peening) to the
unaltered surface finish without peening-induced defects. Mannava et al. (2011) applied
laser peening to orthopedic implants to restore the fatigue strength caused by machining
instabilities. They proved that the rigid spinal implant rod can be modified for flexibility
and laser shock peened to increase the fatigue strength. This enhancement will enable the
use of the implant for longer periods and higher loads in surgical processes with or
International Journal of Structural
Integrity
Vol. 6 No. 1, 2015
pp. 107-123
© Emerald Group Publishing Limited
1757-9864
DOI 10.1108/IJSI-06-2014-0028
IJSI
6,1
Downloaded by University of Louisville At 10:01 05 February 2015 (PT)
108
without fusion. This technology can be readily applied to all metals that are certified to be
used for human implant, so the need for clinical trials is minimized. Lim et al. (2012)
showed that laser peening is a practical option for improving abrasion and corrosion
properties of seawater desalination pump parts. They found that the size and number of
corrosion pits can be decreased by half as a result of laser peening process. Although
conventional laser peening process takes advantage of the sacrificial overlay, laser
peening without coating has been employed for the underwater maintenance of nuclear
power plants in Japan (Sano et al., 2006). This process results in surface roughness and
may have thermal effects due to direct ablation, but can increase in the fatigue life of the
component. The method’s advantages in the application in remote environments such as
nuclear facilities make it a preferable choice for certain operations.
The schematic representation of a laser peening process is shown in Figure 1.
A higher energy laser source emits laser pulse (GPa) striking the component surface
and resulting in the generation of plasma. This plasma expands and creates a high
magnitude short duration pulse which is propagated into the material as a shock wave
capable of plastically deforming the material near the surface regions. The elastically
deformed region surrounding the plastically deformed zone tries to come back to its
initial state, resulting in the formation of CRS. Researchers have conducted extensive
investigations to comprehend the shock wave physics taking place during the
formation of the laser-induced plasma (Fabbro et al., 1990). The transparent overlay
confines the plasma from rapid expansion away from the material surface which
helps to obtain a high intensity pressure pulse into the material. Black paint or tape is
considered most of the time as the opaque overlay while flowing water along the
peening surface is usually taken as the transparent overlay.
Numerical simulations such as finite element analysis (FEA) are very useful tool
to predict the optimum processing conditions to obtain the most favorable outcome
without the need of conducting many costly experiments. Braisted and Brockman
(1999) created two dimensional axi-symmetric models using FEA to predict the residual
stresses generated during a single-shot laser peening process. Arif (2003) used finite
difference algorithm to simulate the stress wave propagation and combined it with
finite element module to predict the deformations and residual stresses. Hu et al. (2006)
Transparent
Overlay
Component
Laser
Source
Laser Pulse
Plasma
Generation
Figure 1.
Schematic of laser
peening process
Opaque
Overlay
Shockwaves
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conducted 3D laser peening simulations and they obtained good correlation with
experimental data. They also constructed a symmetric cell with overlapped laser shots
that can be used for simulating large-scale laser peening problems efficiently by
duplicating the residual stresses predicted in a single cell to regions surrounding the
cell (Hu and Yao, 2008). Singh and Grandhi (2010) developed an advanced strategy by
creating surrogate models from multi-fidelity simulation to conduct parametric
optimization of laser peening design variables. Spradlin et al. (2011) developed a
framework for fatigue life estimation of laser peened components and validated it
with experiments.
Although many efforts have been put into understanding the shock wave
phenomenon on flat components, predicting the effects of LP on curved components
(which represent the majority of realistic problems) poses a huge challenge. Yang et al.
(2008) conducted two-sided laser peening for aluminum alloy rods and showed that
geometric effects influenced the residual stress field significantly. Vasu and Grandhi
(2013) observed that concave geometries had higher residual stress compared to
convex geometries under similar input conditions. This research constitutes an attempt
to relate the residual stress profiles based on the plasticity incurred in the material for
curved geometries involving convex and concave nature as well as cylindrical and
spherical curvatures. It is found that the release waves generated from the boundary of
the impact of the laser shot result in different plasticity between curved geometries and
flat geometries (Vasu et al., 2013). These differences in plasticity lead to the differences
observed in the residual stresses for the curved geometries. An analytical relationship
is derived for laser energy as a function of the curvature of the material for each
geometry. A near-linear relationship is observed for both the simulation and analytical
results. The spherical models have higher CRS in a concave geometry and a lower CRS
in a convex geometry when compared to cylindrical models.
2. 3D numerical modeling and analysis of laser peening by FEA
The residual stress distribution of laser peening process for curved geometries is
numerically predicted by FEA. The FEA model consists of finite elements depicting the
deformation zone, while the boundaries are represented using non-reflective infinite
elements as shown in Figure 2. Finite elements are composed of C3D8R elements that
represent continuum, three-dimensional, eight-node, and reduced integration elements
(green region in Figure 2). Infinite elements are assumed to be elastic elements
comprising eight-node linear CIN3D8 elements (red region in Figure 2). While modeling
and meshing the entire component is computationally expensive, laser peening affects
only a small region of the component. Therefore modeling the critical locations as
a combination of finite and infinite elements helps achieve efficiency for the laser
peening simulation problem. A single laser shot with 2.5 mm radius striking the surface
of the material is modeled using a quarter-symmetric configuration for computational
efficiency. The dynamic loading utilized for this analysis has a sharp, Gaussian,
temporal profile, and a uniformly distributed spatial profile with a peak pressure of 5.5
GPa (Vasu et al., 2013). In total, two convergence studies are performed for all selected
geometries; mesh convergence to inspect the mesh stability, and domain convergence
to examine whether the simulation domain can capture the plastically affected zone.
Based on the results, a 40 μm element size and a 6 × 6 × 6 mm simulation domain are
chosen (3,375,000 finite elements and 67,500 infinite elements). The modified explicit
procedure has been utilized in this research. It approximates the residual stress field by
using only an explicit solver instead of a combination of explicit and implicit solvers to
Plasticity
generated by
Rayleigh
waves
109
IJSI
6,1
Infinite
Elements
Finite Elements
Laser Spot
110
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Symmetry
Symmetry
Figure 2.
Quarter-symmetric
FEA model for a
flat geometry
reach the equilibrated state (Vasu et al., 2013). The ability to predict similar results to
those of the explicit-implicit procedure makes the modified-explicit procedure appealing
to be used for laser peening simulations because it requires considerably less time.
LP is a high strain rate process. The material response of the component to high
strain rates can be significantly different from static and quasi-static loading conditions.
Hence, an accurate constitutive model has to be chosen to simulate the material behavior.
The Johnson-Cook model generates results that agree with the experimental ones, and is
adopted in this paper for modeling laser peening processes (Amarchinta et al., 2010). Since
the thermal effect is minimal for LP processes, the temperature term can be removed from
the model, and hence the equivalent Von Mises stress σ is given by:
h
i (1)
s ¼ A þ Benp 1 þ Cln_e n
where εp is the equivalent plastic strain, e_ n ¼ e_ =_e 0 is the dimensionless strain rate,
e_ represents the strain rate from the high strain rate experiments, and e_ 0 is the reference
strain rate. A, B, C, and n are the material constants. Constant A is the yield stress at
0.2 percent offset strain; constant B, and exponent n represent the strain hardening effect.
The expression in the second bracket represents the strain rate effect through constant C.
Ti-6Al-4V is the material considered throughout this investigation. The material model
input parameters for the simulation are tabulated in Table I (Vasu et al., 2013).
2.1. Definition of 3D curved geometries
The quarter-symmetric models are constructed using spherical (r, θ, ϕ) and cylindrical
co-ordinates (r, θ, z) to describe the convex and concave curvatures for spherical
and cylindrical geometries, respectively. The FEA models are constructed as shown
Table I.
Johnson-cook
material model
constants for
simulation
Material
Ti-6-Al-4V
A (MPa)
B (MPa)
n
C
e_ 0 (/s)
E (GPa)
n
ρ (kg/m3)
950.2
603.4
0.1992
0.0198
0.0009
113.8
0.342
4,500
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in Figure 2. A condition of θ ¼ ϕ is utilized for modeling the double curvatures of the
spherical geometry where θ denotes the polar angle and ϕ is the azimuth angle. The
curved geometries for this research are defined by the component curvature (θ), which
has inverse relationship with radius of curvature (r) of the geometry. The arc lengths (l)
of all the FEA models are maintained the same for all geometries. The applied pressure
is assumed to be acting normal to the material surface. The presence of curvatures
makes the prediction with individual stress components unsuitable for comparison.
Hence minimum principal stress characterizing the maximum compression on the
material is used as the measure of the CRS imparted to the component (Figure 3).
Plasticity
generated by
Rayleigh
waves
111
2.2. Numerical simulation results
The residual stress predictions for cylindrical models along the curved edge on the
surface (X-direction) are shown in Figure 4(a). A specific trend can be noticed:
CRSConvex o CRSFlat o CRSConcave. Even for a non-uniform spatial spot profile, similar
trends are detected (Vasu and Grandhi, 2013). As we increase the curvature in a convex
model, the CRS decreases. An increase of curvature in a concave model results in an
increase of the CRS. In the Y-direction (depth), similar trends in the near-surface regions
can be perceived. However, in the Z-direction (flat side, no curvature), no special trend is
observed (Vasu et al., 2013). The results indicate that, the curvature effects are
dominant along the curved side of the cylindrical model in comparison with the
flat side. Also, the curvature follows a near-linear relationship with the average
CRS observed on the surface of the peened component. The relationship is shown
in Figure 4(b).
The CRS prediction for spherical models along the X-direction is exhibited in
Figure 5(a). It is observed that the CRS generated on the surface in orthogonal directions
(X and Z-direction) are the same, which is expected due to symmetric nature of the
quarter-symmetric spherical model. Unlike the cylindrical model, all three directions
(a)
Convex
Surface
Concave
Surface
Z
X
X
Y
Z
Y
(b)
Concave
Surface
Convex
Surface
Z
X
Y
Notes: (a) Cylindrical models; (b) spherical models
X
Z
Y
Figure 3.
Curved FEA models
IJSI
6,1
(a)
200
CRSConvex < CRSFlat < CRSConcave
112
CRS (MPa)
0
–200
–400
–600
Flat, =0
–800
Convex, =90
Concave, =90
=Curvature
–1,000
0
2
1
3
4
5
Distance along X (mm)
(b)
–750
X
–800
Avg. CRS (MPa)
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–1,200
–850
Convex Curvature
Concave Curvature
–900
X
–950
–1,000
Figure 4.
CRS distribution for
cylindrical models
Z
Y
0
Z
Y
20
40
60
80
100
, Curvature (deg)
Notes: (a) Curvature effect; (b) linearity
show the same trend. However, spherical models also exhibit the linearity similar to
cylindrical models. The near-linear relationship is shown in Figure 5(b).
The comparison of CRS prediction of spherical models to cylindrical models is
shown in Figure 6. It can be noticed that the slope of the spherical geometry is steeper
than the slope of the cylindrical geometry creating an “enveloping effect,” where the
spherical geometry is enveloping the cylindrical geometry. The difference in CRS is due
to the difference in plasticity generated by the shockwaves. This will be discussed in
the next section.
3. Difference in plasticity for curved geometries
The vaporization of the material due to the impact of intense laser radiation causes the
formation of plasma. The presence of the tamping layer (generally water) prevents
the plasma from expanding from the material, resulting in the formation of a pressure
pulse propagated into the material as shockwaves. The impact of LP-induced
shockwaves inside the material can be explained as a two-step process as shown in
(a)
200
Plasticity
generated by
Rayleigh
waves
CRSConvex < CRSFlat < CRSConcave
0
CRS (MPa)
–200
–400
Flat, = 0
–800
Convex, =90
Concave, = 90
= Curvature
–1,000
–1,200
2
1
0
3
4
5
Distance along X (mm)
(b)
–700
Avg. CRS (MPa)
–750
X
Z
Y
–800
Convex Curvature
Concave Curvature
–850
–900
–950
–1,000
Z
Y
X
0
20
40
60
80
100
, Curvature (deg)
Notes: (a) Curvature effect; (b) linearity
Figure 5.
CRS distribution for
spherical models
–700
Convex Spherical
Concave Spherical
Convex Cylindrical
Concave Cylindrical
–750
Avg. CRS (MPa)
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113
–600
–800
–850
–900
Flat
–950
0
20
40
60
, Curvature (deg)
80
100
Figure 6.
Effect of cylindrical
and spherical
geometries on
component curvature
IJSI
6,1
114
Figure 7. In the first step, plane waves are generated and propagated in the normal
direction of the material generating plastic deformation along the contact surface regions
leading to a material discontinuity along the laser spot periphery. In the second step,
the discontinuity caused by the plane waves leads to the formation of two release waves,
P-wave and S-wave, which travel along the longitudinal and transverse directions,
respectively (Forget et al., 1993). The release wave in the transverse direction (S-wave)
interacts with the surface forming a Rayleigh wave. This Rayleigh has a retrograde
motion with elliptical trajectory acting against the plastic deformation created by the
plane waves (Aki and Richards, 2002). A reduction in plasticity ensues due to this
phenomenon. This phenomenon, termed as reduced plasticity (RP), may lead to a
reduction in CRS and, in some cases, form unfavorable tensile residual stresses on the
surface of the peened component.
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3.1. RP on flat, spherical, and cylindrical geometries
The time history plots from the simulation as shown in Figure 8 demonstrate the
reduction in plastic deformation observed in the structural components. The sequence
of events can be observed: the initial plastic deformation caused by plane waves and
the second phase of RP phenomenon caused by the Raleigh waves. Figures 8(a) and (b)
compare the plastic strain history for flat, cylindrical, and spherical geometries,
respectively, until a stable strain state is achieved. It can be observed that the initial plastic
deformation caused by the plane waves is nearly the same for all three geometries.
However, the differences start to emerge for all the geometries when the Rayleigh waves
(a)
Plane
Wave
LP
Material
(b)
P=Longitudinal release wave
S=Transverse release wave
R= Particle motion of Rayleigh wave
S
S
P
P
R
Figure 7.
Shock wave
propagation in
laser peening
Material
Notes: (a) Plane waves; (b) release waves
(a)
Plasticity
generated by
Rayleigh
waves
Flat
Convex
Concave
–0.005
115
RPCylindrical
–0.01
–0.015
–0.02
0
0.1
0.2
0.3
0.5
0.4
Time (s)
(b)
0
Minimum Principal Strain
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Minimum Principal Strain
0
Flat
Convex
Concave
–0.005
RPSpherical
–0.01
–0.015
–0.02
0
0.1
0.2
0.3
Figure 8.
RP in curved
geometries
0.5
0.4
Time (s)
Notes: (a) Cylindrical; (b) spherical
reach the specified location. In this case, the strains are measured 1.5 mm away from the
center of the spot for all the geometries. Maximum RP occurs in the convex geometry
while the minimum occurs in the concave geometry. Since higher plastic strain is a direct
indicator of higher residual stress, the convex model has the least CRS while the concave
model has the highest CRS. Figure 8 also shows the enveloping effect of plastic strain by
the spherical geometry over the cylindrical geometry. The peak pressure is determined to
be a significant factor in creating the differences in plasticity for the various geometries
(Vasu et al., 2013).
3.2. Plastic dissipation energy for curved geometries
The energy lost due to plastic deformation can be calculated from the simulation by
means of plastic dissipation energy. It is expressed as:
Z t Z
Plastic dissipation energy ¼
s e_ dV dt
c pl
0
V
(2)
IJSI
6,1
116
where σc is the stress calculated from Johnson-Cook Model, e_ pl is the plastic strain rate,
V and τ are the variables representing volume and time, respectively. Figures 9 and 10
show the differences in the plastic dissipation energy for cylindrical and spherical
geometries, respectively. Figures 9(a) and 10(a) indicate that the maximum plasticity is
induced in the convex geometry while the minimum plastic deformation is detected in the
concave geometry. The energy is nearly the same for all geometries initially (deformation
due to plane waves) until the release waves come into effect. This difference in plasticity
induced by the release waves is the primary reason for the difference in residual stresses.
It can also be verified that a near-linear relationship exists between the plastic dissipation
energy and the curvature of the material as shown in Figures 9(b) and 10(b), respectively.
Plastic Dissipation Energy (J)
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(a)
0.08
0.06
Flat Model
Convex
Concave
0.04
0.02
0
0
4
2
6
Time (s)
Plastic Dissipation Energy (J)
(b)
0.074
0.072
0.068
0.066
0.064 0
Figure 9.
Plastic deformation
in cylindrical
geometries
Convex Curvature
Concave Curvature
0.07
20
40
60
80
, Curvature (deg)
Notes: (a) Plastic dissipation energy vs time; (b) plastic
dissipation energy vs curvature
100
(a)
Plasticity
generated by
Rayleigh
waves
0.06
Flat
Convex
Concave
0.04
117
0.02
0
0
2
6
4
Time (s)
(b)
0.08
Plastic Dissipation Energy (J)
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Plastic Dissipation Energy (J)
0.08
0.075
Convex Curvature
Concave Curvature
0.07
0.065
0.06
0.055
0
20
40
60
80
100
, Curvature (deg)
Notes: (a) Plastic dissipation energy vs time; (b) plastic
dissipation energy vs curvature
4. Analytical formulation for the energy density (ED) of structural
geometries
We assume that the laser energy (E) imparted onto the material is same for the flat and
curved geometries. However, the plastically affected volumes (PAVs) for these
geometries are different, which is due to observable variations in the plastic dissipation
energy as explained in Section 3.2. Since the energy dissipates as the shock waves go
through the material, a finite volume is significantly affected due to the plastic
deformation near the surface regions. Therefore, the ED, defined as laser energy per
unit volume is utilized in this research as a measure of plastic deformation, is a function
of PAV only. For a laser spot of 5 mm diameter (representing the arc length, lc), let us
assume that it creates plastic deformation up to a depth d. Plane waves generate similar
plastic deformation in all three geometries as observed in Section 3.1. In this section,
PAV’s for flat, spherical, and cylindrical geometries are derived to show the difference
in plasticity. Figure 11 shows a schematic representation (axi-symmetric) of the
Figure 10.
Plastic deformation
in spherical
geometries
IJSI
6,1
d
118
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r = ∞, c = 0
Flat
Model
Figure 11.
PAV of a flat
geometry
plastically affected area for a flat geometry. The PAV of a flat geometry can be
represented in cylindrical co-ordinates as:
Z rZ
2p
Z
d
dV ¼
RdRdydz
0
0
(3)
0
Hence the PAV for the flat geometry is calculated as:
V Flat ¼
pl 2c d
4
(4)
4.1. Derivation of the PAV for spherical geometries
Figure 12 shows axi-symmetric representation of the PAV’s for spherical geometries
with concave (Figure 12(a)) and convex curvatures (Figure 12(b)). The differential PAV
of curved geometries can be represented in spherical co-ordinates as:
Z rZ
yc =2
dV ¼
0
0
Z
2p
R2 SinfdRdfdy
(5)
0
Therefore, the volume for the concave and convex geometries for the spherical models
can be calculated as:
2p
yc
1Cos
V Concave ¼
(6a)
ðr þ dÞ3 r 3
3
2
(a)
(b)
Plasticity
generated by
Rayleigh
waves
d
r
c /2
119
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d
c /2
r
Figure 12.
Plastically affected
volumes in spherical
geometries
Notes: (a) Concave; (b) Convex
V Convex ¼
2p
yc
1Cos
r 3 ðrdÞ3
3
2
(6b)
where θc is the curvature of the plastically affected zone in degrees, r is radius of
curvature in mm, d is plastically affected depth in mm, and lc is arc length in mm. It can
be mathematically proven that as r→N, θc→0, VFlat ¼ VConcave ¼ VConvex. Since arc
length (lc) is constant for this problem (the arc length is equal to the diameter of the laser
spot for each geometry), the PAV can be considered to be a function of curvature alone
with the formula rθc ¼ lc. Therefore, we obtain:
3 3 !
2p
yc
lc
lc
1Cos
þd V Concave ¼
(7a)
3
2
yc
yc
V Convex
3 !
3 2p
yc
lc
lc
1Cos
¼
d
3
2
yc
yc
(7b)
4.2. Derivation of the ED for cylindrical geometries
For a cylindrical model, an equivalent PAV is created for the analytical formulation
because of the complicated shape. Figures 13(a) and (b) show the equivalent PAV’s for
concave and convex geometries, respectively. The effect of the PAV is approximated
by a circular top area indicating the laser spot surface (red) and an elliptical bottom
area (black). So the PAV can be calculated by the formula:
Z
d
V Concave ¼
parðxÞdx
0
(8)
IJSI
6,1
(a)
2acv
dx
FRONT
VIEW
d
SIDE
VIEW
120
2bcv
bcv
acv
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TOP
VIEW
(b)
2acx
dx
FRONT
VIEW
d
SIDE
VIEW
2bcx
bcx
Figure 13.
Equivalent PAV’s in
spherical geometries
acx
TOP
VIEW
Notes: (a) Concave; (b) Convex
where r(x) for the concave and convex geometries is calculated by Equations (9a) and
(9b), respectively:
bcv acv
x
(9a)
r cv ðxÞ ¼ acv þ
d
bcx acx
x
(9b)
r cx ðxÞ ¼ acx þ
d
Since acv ¼ acx ¼ lc/2, bcv ¼ lc(r+d)/2r, bcx ¼ lc(r-d)/2r, and rθc ¼ lc, the PAV for a
cylindrical model is calculated as a function of θc alone and is represented by the
Equations (10a) and (10b) for the concave and convex curvatures, respectively:
pl 2c d 2 þ yc d
8
lc
2 pl c y 2yc d
¼
8
lc
V Concave ¼
(10a)
V Convex
(10b)
5. Discussion
Since the input laser energy is assumed to be a constant, curvature is the only variable
required to calculate the ED. From the calculation of PAV’s for the various geometries
represented in Section 4, the ED is non-linear. However, we can see that there is a nearlinear relationship between the CRS and curvature from the numerical simulation
results discussed in Section 2.2. It should be noted that FE simulation considers the
curvature for the entire FE model (θ), not just the curvature of the plastically affected
zone (θc). The relationship between θ and θc is given by:
lyc
lc
121
(11)
Since most of the practical problems will have curvatures (θ) less than 90°, the
relationship between ED and curvature needs to be considered between 0° (flat
geometry, infinite radius of curvature) and 90°. The ED is plotted against curvature as
shown in Figure 14 (initial conditions are E ¼ 80 J, d ¼ 1 mm, lc ¼ 5 mm, l ¼ 12 mm), and
it can be observed that a near-linear relationship exists between them although the
functional relationship is non-linear.
To explain the reason behind the linear relationship from a non-linear function, the
PAV for the spherical geometry can be broken down into a much simpler function by
using cubic expansion formula and Taylor series expansion of cosines. The resultant
relationship for the ED is given by:
EDConcave ¼
EDConcave ¼
2p
3
2p
3
1 cos
1 cos
yc
2
yc
2
E
E
r 3 ðrd Þ3
ðr þ d Þ3 þ r 3
E=4pdl 2c
1 dyl c c
E=4pdl 2c
1 þ dyl c c
(12a)
(12b)
The non-linear functions for the concave and convex geometries given in Equations
(12a) and (12b) are in the forms of c/(1−x) and c/(1+x), respectively, where c ¼ E=4pdl 2c ,
5
Energy Density (J/mm3)
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y¼
Plasticity
generated by
Rayleigh
waves
Convex Spherical
Concave Spherical
Convex Cylindrical
Concave Cylindrical
4.8
4.6
4.4
4.2
4
3.8
3.6
Flat
3.4
0
10
20
30
40
50
60
70
, Curvature (deg)
80
90
100
Figure 14.
Energy density vs
curvature
IJSI
6,1
and x ¼ dyc =l c ¼ pyc d=180‘c . Applying another Taylor series expansion at the origin,
an infinite Taylor series is obtained for the non-linear functions:
EDConcave ðxÞ ¼ c 1 þ x þ x2 þ x3 þ x4 þ :::
(13a)
EDConvex ðxÞ ¼ c 1x þ x x þ x :::
(13b)
2
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122
3
4
For θ ¼ 90°, l ¼ 12 mm, and lc ¼ 5 mm, θc is determined to be 37.5° for convex and
concave geometries. Therefore, the variable “x” for this problem is calculated to be
equal to 0.13. Truncating this series to first order, we obtain EDConcave(x)≈c(1+x)
¼ 1.13c. The percentage error is equal to 1.74 percent. Therefore when x is small,
c=1x cð1 þ xÞ. Similarly, for convex geometry, EDConvex ðxÞ ¼ c=1 þ x cð1xÞ
where percentage error is equal to 1.69 percent. Since x is small, we can approximate
the non-linear relationship to a linear one without losing much accuracy. Therefore, the
CRS can be considered to have a linear relationship with the component’s curvature for
a spherical geometry. The same way, it can be proven that a linear relationship exists
for the cylindrical geometry as well.
6. Conclusions
A three-dimensional explicit numerical model is created to predict the residual stress
profiles for a single-shot laser peening process on flat, spherical, and cylindrical
geometries. The results reveal that the CRS follows a near-linear relationship with
respect to curvature and the slope of the CRS curve for spherical geometry is steeper
than the cylindrical geometry. These differences in CRS result from the differences in
the reduction of the amount of plastic deformation for various curvatures. It is found
that the Rayleigh wave originating from the boundary of impact due to the material
discontinuity is the primary cause of the “RP” phenomenon; they impact the initial
plastic deformation created by the plane waves. The linearity shown by the plastic
dissipation energy with the curvature supports the trend shown by the CRS. Analytical
formulations are derived for the ED of flat and curved geometries as a function of
curvature alone. Although the functions are non-linear in nature, a near-linear
relationship is observed given the operating conditions of the curvature. A steeper
slope is observed for spherical geometry, creating an “enveloping effect” over
cylindrical geometry.
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Corresponding author
Dr Anoop Vasu can be contacted at: anoop1984@gmail.com
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Plasticity
generated by
Rayleigh
waves
123